Magnesium Boride Nanotubes: Relative Stability and Atomic and Electronic Structure

Pavel B. Sorokin,*,†,‡,| Leonid A. Chernozatonskii,| Pavel V. Avramov,†,§ andBoris I. Yakobson‡

Siberian Federal UniVersity, 79 SVobodny AVenue, Krasnoyarsk, 660041 Russian Federation, Department ofMechanical Engineering & Material Science and Department of Chemistry, Rice UniVersity, Houston,Texas 77251, Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, 4 Kosigina Street,Moscow, 119334, Russian Federation, and Kirensky Institute of Physics, Russian Academy of Sciences,Akademgorodok, Krasnoyarsk, 660036 Russian Federation

ReceiVed: NoVember 25, 2009; ReVised Manuscript ReceiVed: February 5, 2010

A comparative study of the energies and the electronic structure of MgBx nanotubes is performed within theframework of the density functional theory. Different basic compositions (x ) 2 for diboride and x ) 3 fortriboride) and different diameters (3 Å < D < 18 Å), as well the exterior, interior, and staggered placementof magnesium atoms, are considered. Energy analysis reveals a nontrivial bending behavior of the MgB2

sheets, such that the tubes with exterior and staggered configurations display the energy minima at certainsmall diameters (of the boron cage sublattice). The semiconducting behavior of narrow MgB2 nanotubes withexterior Mg position was observed.

1. Introduction

Magnesium diboride based materials have attracted a lot ofattention because of their remarkable physical properties andwide possible technological applications. Special interest in thecrystalline MgB2 has ignited after discovery of the magnesiumdiboride superconductivity at 39 K.1 The MgB2 nanotubes areexpected to display the same remarkable physical propertiescaused by layered atomic structure of the parent material, bulkmagnesium diboride.

Using simple classical analysis, the existence of MgB2

nanotubes of two isomorphic configurations has been suggested,with interior and exterior magnesium layers relative to the boronlattice. The energies of armchair and zigzag tubes wereapproximately evaluated,2 showing that the armchair nanotubesof the first configuration with interior magnesium layer areenergetically preferable. On the other hand, the paper by Lauet al.3 reported a single narrow zigzag MgB2 nanotube, whilefocusing on different concentrations of magnesium atoms (allexterior) on a boron cage, concluding that the Mg tends todistribute uniformly.

Further, the electronic structure of selected four armchairMgB2 nanotubes of both configurations4 was studied using thedensity functional tight binding (DFTB) method. The DFTBapproach confirmed the energetic stability of the first configu-ration with interior magnesium layer. Another study5 reportedDFT calculations of a MgB2 nanotube with the exterior (withrespect to the boron cage) magnesium layer. The crystallinetubular structure based on the MgB2 nanotubes has also beenstudied using DFT calculations.3,6 Overall, it was shown thatall MgB2 nanotubes and related structures display metallicproperties.3–6

In another line of recent research of the most favorable boronatom arrangement in the pure-boron hollow clusters (boronfullerenes7–11 and nanotubes12–14), the original Aufbau principlefor boron clusters15,16 was modified.7 It was found that theperiodic exclusion of boron atoms from the uniform triangularboron lattice of hollow clusters leads to improved energeticstability.17 In the case of curved clusters, the effect can beadditionally interpreted in the following way. The curvature ofa cluster increases the tension of the atomic structure, and aninsertion of empty hexagons (or removal of some boron atoms)decreases it. Along with this, on the other hand, the deletion ofselected boron atoms deviates from the Aufbau principle andthus decreases the stability of the structure. The “golden mean”in the competition of both effects is the 1/3 ratio between thenumber of empty and filled boron hexagons/pentagons in theatomic geometry of the boron clusters.7 One can similarlysuggest that the modified Aufbau principle7 can also be appliedto other boron-based nanoclusters, e.g., in transforming MgB2

into MgB3 nanotubes.Motivated initially by this possibility of varying the composi-

tion and structure-property relationship of MgBx (x > 0)nanotubes, we systematically designed and studied the atomicmakeup and electronic properties of MgB2 and MgB3 composi-tions in different isomorphic configurations. It becomes ofparticular interest due to the recent report of microscopyobservations of the large diameter MgB2 nanotubes and theirsuperconductivity.18 The experimental observations of othermetal boride (Fe-B, Co-B, and Ni-B) nanotubes19 furthersupport the feasibility of the MgBx family as well.

2. Method and Model

The calculations of the atomic and electronic structure ofmagnesium diboride and tridoride nanotubes (MgB2-NT andMgB3-NT) were performed using density functional theory20,21

within the local density approximation for the exchange-cor-relation functional,22 employing norm-conserving Troullier-Martins pseudopotentials23 in the Kleinman-Bylander factorizedform.24 Finite-range numerical pseudoatomic wave functions

* Corresponding author. E-mail: PSorokin@iph.krasn.ru.† Siberian Federal University.‡ Rice University.| Emanuel Institute of Biochemical Physics, Russian Academy of

Sciences.§ Kirensky Institute of Physics, Russian Academy of Sciences.

J. Phys. Chem. C 2010, 114, 4852–48564852

10.1021/jp9112014 2010 American Chemical SocietyPublished on Web 03/02/2010

were used as an atomic orbital basis set. Nanotubes were treatedin a tetragonal supercell scheme allowing enough empty spacebetween them to make intermolecular interactions negligible.The geometry of the structures was optimized until residualforces became less than 0.04 eV/Å. The real-space mesh cutoffwas set higher than 125 Ry. The Monkhorst-Pack25 specialk-point scheme was used with 0.1 Å-1 k-point spacing. We used

the SIESTA package26,27 in all calculations. All the values givenabove were carefully tested and found optimal.

3. Results and Discussion

3.1. Atomic Structure. The MgB2 nanotubes consist ofseveral concentric cylinders formed by rolled hexagonal mag-nesium and boron graphite-like plane networks. Alternatively,

Figure 1. Atomic structure of (6, 6) MgB2 armchair nanotubes with interior (a), staggered (b), and exterior (c) Mg layers. MgB2 crystalline lattice,with highlighted atoms isomorphic to the corresponding nanotubes, is presented on the right-hand side of the figure. (d) Structure of {6, 0} MgB3

nanotube and the perspective view of a MgB3 single layer. The unit cells of the tubes are marked by dotted rectangles.

Figure 2. Dependence of the relative (Erel) energy of magnesium diboride nanotubes upon the diameter (D) of corresponding boron cages. Theenergy dependence of the tubes with interior Mg configuration is presented as a green line with solid triangles. The energy dependence of the tubeswith staggered Mg configuration is presented as a red line with empty circles. The energy dependence of the tubes with exterior Mg configurationis presented as a blue line with solid circles. The energies of a single MgB2 layer with exterior and staggered magnesium layer is presented as blueand red horizontal solid lines, respectively. In the insets the dependences of relative energies of magnesium diboride nanotubes upon inversediameter (MgB2-NT with interior magnesium layer) and inverse diameter in the third power (MgB2-NT with exterior and staggered magnesiumlayer) are shown.

Stability and Structure of MgB2 Nanotubes J. Phys. Chem. C, Vol. 114, No. 11, 2010 4853

one can view the atomic structure of MgB2 nanotubes as a boronhexagonal framework with magnesium atoms in the centers ofthe hexagons.2 First predictions of the metal boride nanotubeswith such structure have been made concurrently by Quandt etal.28 (for AlB2) and Chernozatonskii2 (for MgB2).

We studied three different configurations of magnesium ionarrangement relative to the boron framework: interior (withinterior magnesium layer, Figure 1a), intermediate staggered(with interior and exterior magnesium layers, Figure 1b), andexterior (with exterior magnesium layer, Figure 1c) configura-tions. To clarify the structure of the tubes, we also present (onthe right of Figure 1) the atomic lattice of the bulk MgB2 andhighlight the atoms involved in the corresponding nanotubeconstructions. We investigated only armchair tubes because ourpreliminary calculations and the data from refs 2 and 4 giveevidence that these are most energetically favorable structures.

To introduce classification of MgB3 nanotubes, let us followthe classification of boron tubes.12–14 It is necessary to note thatthe basis vectors of the precursor boron plane are turned 30°relative to the corresponding hexagonal graphene lattice. Fol-lowing classification12–14 the MgB3 nanotubes {N, M} can bedivided into three classes: (i) zigzag nanotubes with N ) M,(ii) armchair nanotubes with N * 0 and M ) 0, and (iii) chiralnanotubes with N * M. As an example the atomic structure of{6, 0} MgB3 armchair nanotube is presented in Figure 1d.

Strictly speaking, the MgB2 nanotubes with staggered mag-nesium layer have different classification from MgB2-NT withexterior and interior magnesium layer because of the twicebigger lattice vectors of the basis layer of the tubes. But forcomparison we use the common classification for all MgB2-NT.

3.2. Energetic Properties. To evaluate relative stability, itis convenient to compare the energy of a MgB2 nanotube (Figure2) with the energy of bulk MgB2 material, per atom (essentially,the formation energy of a tube from bulk crystal)

where Etube refers to the tube and Ebulk is the energy of bulkmagnesium diboride. Choosing the energies per atom rather thanper formula unit has an advantage when one wants to comparedifferent compositions, such as MgB2 and MgB3 in the presentcase.

With increasing diameter D, the energies of MgB2 tubes withexterior and staggered magnesium layers approach the energyof planar MgB2 sheet, and each has a minimum at a certain Dof the boron cages, in contrast to the carbon single-wallnanotubes, for which E(D) ∼ 1/D2 is monotonous.29 The energydependence can be formally approximated by the E(D) )A((1/D3) - (1/D0

3))2 - B. In the case of MgB2-NT with exteriormagnesium layer, A ) 230 (eV ·Å6/atom) and B ) 0.80 eV/atom. The minimum D0 of the curve corresponds to the favorablestructure and is equal to 4.2 Å, which is close to the diameterof the (3, 3) tube (4.0 Å). For MgB2-NT with staggeredmagnesium layer, A ) 41 ·103 (eV ·Å6/atom) and B ) 0.81 eV/atom with D0 equal to 9.4 Å, which is close to the 9.8 Ådiameter of the (6, 6) tube. (The accuracy of the fittedcoefficients is based on adjusted R2 > 0.98 in all cases.)

The energy dependences of the tubes tend to the energies ofcorresponding MgB2 sheets (E(∞) of MgB2-NT with exteriorconfiguration is equal to 0.84 eV/atom; E(∞) of MgB2-NT withstaggered configuration is a little bit larger and is equal to 0.87eV/atom). The nonmonotonic energy dependence behavior wasalso found for SiO2,30,31 C2F,32 and SiH33 nanotubes. Theenergies of (n, n) tubes with exterior magnesium layers for n >9 are lower than the corresponding energies of the tubes withstaggered magnesium layers. This result means that MgB2 tubeswith diameters D < 8 Å and D > 14 Å more likely containmagnesium layer with exterior configuration, whereas tubes withdiameters 8 Å < D < 14 Å would rather contain magnesiumlayer with staggered configuration.

The planar MgB2 sheet is less stable than most tubes withexterior and staggered magnesium layers. Such low stability ofsingle MgB2 sheet was predicted by Chernozatonskii2 and Saitoet al.5 The atomic sheets in graphite are bounded by weak vander Waals forces and can be roughly treated as independent,whereas the ionic interlayer interaction in bulk MgB2 is muchstronger.5 Therefore, it is possible to obtain the single layer ofgraphite,34 whereas obtaining a single flat layer of MgB2 seemsto be unrealistic.

Mg-Mg interactions in the tubes make the main contributionto the stability of the staggered configuration. Distances betweenneighboring magnesium ions in both interior and exterior layersin the favorable MgB2 (6, 6)-NT are close to the bulk value of3 Å,1 whereas for tubes with other diameters this value isnoticeably different. The most energetically favorable tubes withexterior Mg layer display Mg-B distances closest to the bulkvalues.

The dependence of the relative energy on diameter D ofMgB2-NT with interior Mg configuration displays monotonicbehavior due to repulsion of interior magnesium ions. Becauseof this, the (n, n) tubes with n < 6 are unstable. The best fit ofthe energy dependence is inversely quadratic E(D) ) A(1/D)2

+ E0, where A ) 41.64 (eV ·Å2/atom) and E0 ) 0.843 (eV/atom) is the strain energy of the single MgB2 sheet. The interiormagnesium ions visibly expand the boron cage, and thereforediameters of the tubes are much bigger than the diameters ofcorresponding tubes with exterior and staggered Mg configura-tions. Our results are different from the DFTB results ofIvanovskaya et al.4 who reported that nanotubes with interiormagnesium layer are energetically more favorable.

Figure 3. Gibbs free energies of formation of the (6, 6) MgB2-NT(red line with empty circles) and {6, 0} MgB3-NT (red line with emptystars) with staggered magnesium layer configuration, plotted as afunction of the boron chemical potential µB (shifted to the value of µB

in the R3jm solid state boron phase). In the inset the similar dependenciesof Gibbs free energies of formation of (3, 3) MgB2-NT (blue line withfilled circles) and {3, 0} MgB3-NT (blue line with filled stars) withexterior magnesium layer configuration are presented.

Erel ) Etube - Ebulk

4854 J. Phys. Chem. C, Vol. 114, No. 11, 2010 Sorokin et al.

The relative stability of MgBx-NT of different stoicheometryx depends of course on the constituents’ chemical potentials,µMg for magnesium and µB for boron, which in turn representenvironmental conditions of synthesis. To account for this effect,we used the Gibbs free formation energy δG35 for the MgBx,as

The dependence of δG(x, µB) on µB for MgBx-NT systems isdepicted in Figure 3. When choosing the µMg and µB, we donot try to suggest specific synthetic conditions (generally adaunting task for theory), but simply select some realistic valuesfor the constituent elements and evaluate the (meta)stability ofMgBx tubes with respect to possible decomposition/segregationinto pure B and pure Mg. To compare the relative stability ofMgBx-NT for different x, the value µMg was fixed and equal tothe chemical potential of bulk magnesium (P 63/mmc). Thevalue µB was chosen to vary due to the existence of a set ofboron allotropes in nature. One can imagine a possible situationwhen MgBx is obtained by reacting a B-containing vaporcomponent with Mg substrate, in a CVD-type approach. In thiscase µB varies by experimental conditions (pressure, temperature,and B-feedstock choice).

The µB parameter can be calculated by the same method forR3jm36 (µB

R3jm) and P4jn237 (µBP4jn2) symmetries of two different

phases of boron.In the case of the µBR3jm, the (6, 6) MgB2-NT is

∼0.26 eV/atom more thermodynamically stable than the {6, 0}MgB3-NT. Another µB value for the P4jn2 (µB

P4jn2 - µBR3jm ) 0.12

eV/atom) leads to decreasing of the difference between the (6,6) MgB2-NT and {6, 0} MgB3-NT energies up to ∼0.25 eV/atom. The MgB3 phase can become more favorable only at µB

> 3.06 eV (Figure 3).3.3. Electronic Structure. Similar to crystalline MgB2, all

examined MgB2 nanotubes, except the (2, 2) (Egap ) 0.63 eV)and (3, 3) (Egap ) 0.45 eV, Figure 4a) MgB2-NT with exterior

Mg arrangement, display metallic properties. Increase of theMgB2 nanotube diameter leads to the closing of the band gapand restoring the metallic behavior. The presence of the nonzeroband gap in MgB2 tubes with diameters D < 5 Å can beattributed to the strain in the atomic structure caused by largenanotube curvature.

4. Conclusions

A comparative study of MgBx nanotubes extends the previousknowledge2–6 and reveals the preferred composition, sublattices(Mg versus B) spatial organization, and diameters. Novel MgB2

nanotubes with staggered magnesium layer configuration arepredicted as a possibility, and the energy change caused bycurvature strain is related to the nanotube diameter. It is foundthat the energy curves of the energetically favorable MgB2 tubeswith exterior and staggered magnesium layer configurations havea pronounced minimum at some preferred diameters of the boroncage. The energy curves of the less favorable MgB2 tubes withinterior magnesium layer are monotonous and rather high. Theenergy dependencies of all tubes tend to the energy of thecorresponding planar MgB2 sheets. MgB2 tubes with exteriormagnesium layer with diameters smaller than 8 Å and biggerthan 14 Å are energetically favorable, whereas tubes withstaggered magnesium layer are favorable with diameters be-tween 8 and 14 Å. It was found that MgB2-NT are generallyenergetically preferable over the MgB3 tubes. All examinedMgB2 nanotubes display metallic properties except very narrowdiameter tubes. The proposed MgB2 nanotubes can be (if theirsynthesis18 is further refined) used in nanoelectronics asinterconnects. Observation of superconductivity in magnesiumdiboride nanotubes18 opens even more tantalizing possibilitiesfor such one-dimensional structures. Further, recent predictionof high hydrogen adsorption on TiB2 nanotubes38 suggests asimilar property possibly for MgB2 structures as well.

Note Added in Proof. After completion of the present study,we became aware of comprehensive comparison39 of various

Figure 4. Electronic band structure of MgB2 nanotubes. (3, 3) MgB2 with exterior, (6, 6) staggered, and (6, 6) interior Mg layer configurations andcorresponding nanotube cross sections. The Fermi level is shifted to zero and marked by the dotted horizontal line.

δG(x, µB) ) Etotal(x) - µMg - xµB

Stability and Structure of MgB2 Nanotubes J. Phys. Chem. C, Vol. 114, No. 11, 2010 4855

geometries of MgB2 layer, which can also be rolled into stableMgB2 nanotubes. This elegant approach can also be applied totest other stoichiometries MgxB1-x, however comparison acrossthe different compositions will depend on the choice ofelemental chemical potentials.

Acknowledgment. P.B.S. and B.I.Y. acknowledge supportby the Basic Energy Sciences division of the Department ofEnergy, award DE-SC0001479. L.A.C. was supported by theRussian Academy of Sciences, program No. 21. P.V.A. andP.B.S. also acknowledge the collaborative RFBR-JSPS grantNo. 09-02-92107-ЯΦ. We are grateful to the Joint Supercom-puter Center of the Russian Academy of Sciences for thepossibility of using a cluster computer for quantum chemicalcalculations. The geometry of all presented structures wasvisualized by commercial ChemCraft software (http://www.chemcraftprog.com).

References and Notes

(1) Nagamatsu, K. J.; Nakagawa, N.; Muranaka, T.; Zenitani, Y.;Akimitsu, J. Nature 2001, 410, 63.

(2) Chernozatonskii, L. A. JETP Lett. 2001, 74, 335.(3) Lau, K. C.; Orlando, R.; Pandey, R. J. Phys.: Condens. Matter

2009, 21, 045304.(4) Ivanovskaya, V. V.; Enjashin, A. N.; Sofronov, A. A.; Makurin,

Yu.N.; Medvedeva, N. I.; Ivanovskii, A. L. J. Mol. Struct. 2003, 625, 9.(5) Saito, S.; Louie, S. G.; Cohen, M. L. J. Phys. Soc. Jpn. 2007, 76,

043707.(6) Prasad, D. L. V. K.; Jemmis, E. D. J. Mol. Struct.: THEOCHEM

2006, 771, 111.(7) Szwacki, N. G.; Sadrzadeh, A.; Yakobson, B. I. Phys. ReV. Lett.

2007, 98, 166804.(8) Zope, R. R.; Baruah, T,; Lau, K. C.; Liu, A. Y.; Pederson, M. R.;

Dunlap, B. I. Phys. ReV. B 2009, 79, 161403.(9) Sadrzadeh, A.; Pupysheva, O. V.; Singh, A. K.; Yakobson, B. I. J.

Phys. Chem. A 2008, 112, 13679.(10) Mukhopadhyay, S.; He, H.; Pandey, R.; Khin, Y.; Boustani, I. J.

Phys.: Conf. Ser. 2009, 176, 012028.(11) Li, M.; Li, Y.; Zhou, Z.; Shen, P.; Chen, Z. Nano Lett. 2009, 9,

1944.(12) Yang, X.; Ding, Y.; Ni, J. Phys. ReV. B 2008, 77, 041402.

(13) Singh, A. K.; Sadrzadeh, A.; Yakobson, B. I. Nano Lett. 2008, 8,1314.

(14) Chernozatonskii, L. A.; Sorokin, P. B.; Yakobson, B. I. JETP Lett.2008, 87, 489.

(15) The Aufbau principle says that highly stable boron nanoclusters,surfaces, and networks can simply be constructed from two basic units only,namely, the pentagonal and hexagonal pyramids, B6 and B7, respectively.

(16) Boustani, I. Phys. ReV. B 1997, 55, 16426.(17) Tang, H.; Ismail-Beigi, S. Phys. ReV. Lett. 2007, 99, 115501.(18) Zhou, S. M.; Wang, P.; Li, S.; Zhang, B.; Gong, H. C.; Zhang,

X. T. Mater. Lett. 2009, 63, 1680.(19) Zhu, Y.; Liu, F.; Ding, W.; Guo, X.; Chen, Y. Angew. Chem. 2006,

118, 7369.(20) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864.(21) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133.(22) Perdew, J. P.; Zunger, A. Phys. ReV. B 1981, 23, 5048.(23) Troullier, N.; Martins, J. L. Phys. ReV. B 1991, 43, 1993.(24) Kleinman, L.; Bylander, D. M. Phys. ReV. Lett. 1982, 48, 1425.(25) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188.(26) Ordejon, P.; Artacho, E.; Soler, J. M. Phys. ReV. B 1996, 53,

R10441.(27) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcıa, A.; Junquera, J.;

Ordejon, P.; Sanchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745.(28) Quandt, A.; Liu, A. Y.; Boustani, I. Phys. ReV. B 2001, 64, 125422.(29) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties

of Carbon Nanotubes; Imperial College Press: London, 1998.(30) Zhao, M.; Zhang, R. Q.; Xia, Y.; Lee, S.-T. Phys. ReV. B 2006,

73, 195412.(31) Chernozatonskii, L. A.; Sorokin, P. B.; Fedorov, A. S. Phys. Solid

State 2006, 48, 2021.(32) Kudin, K. N.; Scuseria, G. E.; Yakobson, B. I. Phys. ReV. B 2001,

64, 235406.(33) Seifert, G.; Kohler, Th.; Urbassek, H. M.; Hernandez, E.; Frauen-

heim, Th. Phys. ReV. B 2001, 63, 193409.(34) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich,

V. V.; Morozov, S. V.; Geim, A. K. Proc. Natl. Acad. Sci. U.S.A. 2005,102, 10451.

(35) Dumitrica, T.; Hua, M.; Yakobson, B. I. Phys. ReV. B 2004, 70,241303.

(36) Decker, B. F.; Kasper, J. S. Acta Crystallogr. 1959, 12, 503.(37) Hoard, J. L.; Geller, S.; Hughes, R. E. J. Am. Chem. Soc. 1951,

73, 1892.(38) Meng, S.; Kaxiras, E.; Zhang, Z. Y. Nano Lett. 2007, 7, 663.(39) Tang, H.; Ismail-Bei gi, S. Phys. ReV. B 2009, 80, 134113.

JP9112014

4856 J. Phys. Chem. C, Vol. 114, No. 11, 2010 Sorokin et al.